A remark on integral geometry. (English) Zbl 1021.53049

Bott, Raoul; Taubes, Cliff

Ghys, Étienne (ed.) et al., Essays on geometry and related topics. Mémoires dédiés à André Haefliger. Vol. 1. Genève: L’Enseignement Mathématique. Monogr. Enseign. Math. 38, 59-83 (2001).

Let \(f=\left\{ f_{1},...,f_{n}\right\} \) be \(n\) homogeneous polynomials in \((n+1)\) variables and let \(\chi(f)\) denote the cardinality of the simultaneous solutions of the equations \(f_{i}=0\), for \(i=1,...,n\), over the reals. In [Computational Algebraic Geometry, Nice 1992, Progr. Math. 109, 267-285 (1993; Zbl 0851.65031)] M. Shub and S. Smale proved that the “average” value of \(\chi(f)\), as \(f_{i}\) varies over the projective space of homogeneous polynomials of degree \(d_{i}\), is equal to \(\sqrt{d_{1}...d_{n}}\).
In this nice paper the authors show how the previous Bezout’s theorem can be derived from a universal integral formula, by introducing a “lifting principle” which allows them to use the push forward of the de Rham theory.
For the entire collection see [Zbl 0988.00114].

Reviewer: Carla Peri (Milano-Largo)

MSC:

53C65

Integral geometry

Keywords:

Bezout’s theorem; integral geometry

Citations:

Zbl 0851.65031